Note: Win Probability changes are based on assuming that both teams are equal in overall talent at any point in time, and that the playing context is neutral between them.

Fast-Forward To Today

I love win probability, win expectancy, or whatever incarnations you know it as. I was first introduced to the concept by the Mills Brothers through the Hidden Game of Baseball. It took a while to find the Mills book, but I did. I posted an excerpt on my site. There are many champions of win probability, including, among others, Doug Drinen, Phil Birnbaum, Dave Studeman, David Appelman, Dan Fox, and Keith Woolner.

It is a very simple concept—what is the chance of winning the game given a certain set of variables? At a minimum, the variables include the inning, score, base, and out. At a maximum, you’d include everything under the sun, such as the identities of the players, the park, the climate, the count, tendencies of managers, and a host of whatever you can think of. Like fan interference. Or managers’ insistence of sticking with a tired pitcher. Or pyschological trauma.

Almost four years ago, I introduced a concept called Leverage Index. Leverage is the swing in the possible change in win probability. If there is a game with one team leading by ten runs, the possible changes in win probability, whether the event is a home run or a double play, will be very close to negligible. That is, there won’t be much swing in any direction.

But, in a late and close game, the change in win probability among the various events will have rather wild swings. With a runner on first, two outs, down by one, and in the bottom of the ninth, the game can hinge on one swing of the bat—a home run and an out will both end the game, but with vastly different outcomes for the teams involved.

You can spot a high-leverage situation, I can spot them, and pretty much everyone can spot many high-leverage situations. All that’s left for us to do is to quantify every single game state into a number. That number is the Leverage Index.

It is important to note that a game state is anything you want it to describe. While I will describe it, for the purposes of this article, as the state of the game given the half-inning, score differential, base, and out situations, you most certainly can expand the game state to include the identities of all the players in the game, the park, the count, and the climate.

I will also be referring to Table 10. Win Expectancy, By Game State from The Book—Playing The Percentages In Baseball. It’s not necessary for you to have this book to follow along. I’ll extract the relevant parts, as needed.

Recipe #1: Absolute Differences

Imagine the easiest possible situations in baseball—you are either safe on a single or you are out. We’ll also assume that a single advances all runners on base by two bases, while the out leaves all the runners unmoved.

Let’s start with a game state of the top of the ninth inning: the home team is ahead by three runs, there are runners on first and second, and there are no outs. The Book tells us the chance of winning the game, for the home team, is 0.841. We have two possible states we can transition into: a safe play puts runners on first and third, and scores a run, for a win. expectancy of 0.701 for the home team, and an out play gives a win expectancy of 0.910. The change in win probability in one case is -0.140, and it is +0.069 in the other case. For our stylized example, we will assume that the single occurs 33% of the time, and the out play occurs 67% of the time.

That spread in possible next states is what specifies the leverage. The fielding/home team gets a gain of .069 wins 67% of the time, and a drop of .140 wins 33% of the time. So, the spread is .069 * .67 + .140 * .33 = .092. That is, the average change in win expectancy for this game state is .092 wins.

We repeat this process for every possible game state, from a score differential of minus 30 runs to plus 30 runs, from the first inning to the thirtieth inning, from bases empty to bases loaded, and for zero, one, and two outs. We also determine how often each of these game states occur. If we multiply the frequency of the game state by the swing for each game state, we end up with an average swing value. In the last few years, the average swing value was .0346 wins. This average swing value becomes our baseline. We divide each of our swing values by the baseline, and we get the Leverage Index (LI).

The average swing value, by definition, will equal to 1.00. If you have an LI of 2.00, this means that this game state has double the leverage than a random game state. That is, an event will have, on average, double the impact at this point in the game, relative to the typical point in the game.

In our example here, we would calculate the LI as .092 / .0346 = 2.7. This makes the situation a high-leverage situation.

This is how it all looks in graph form. The two lower lines represent the typical change in some random point in the game (when the win percentage is .500). The two upper lines represent the change in this specific ninth inning game state (with the win percentage at .841).

Also note that the out lines are thicker, representing the greater frequency than the hit lines.

The gap between the two upper lines is almost three times larger than the gap between the two lower lines. This ratio of almost three times (2.7 to be accurate) is how much more leverage this ninth inning situation has compared to a random situation. This ratio is the Leverage Index.

Recipe #2—Root Mean Square

In terms of calculating the swing value, we can also use a slightly more cumbersome process. You (a) square the swing values, (b) then multiply it by the frequency, (c) sum all the values, and then (d) take the square root. Using these steps, we calculate the swing value for this game state as 0.098. The league baseline would be 0.0418. Our LI calculation becomes .098 / .0418 = 2.3.

Now, we know that baseball is not about just two events. Let’s make it a bit tougher, and introduce a third event: the home run.

Repeating with the above game state, a home run makes this a tie game, in the top of the ninth inning, with no outs. The win expectancy has now plummetted to 0.500, or a drop of 0.341 wins. Let’s also change our frequencies around a little—the home run occurs 3% of the time, the single occurs 27% of the time, and the out occurs 70% of the time. Let’s work out our numbers. Using recipe #1, the swing value is .096 wins, for an LI of 2.8. Using recipe #2, the swing value is .110 wins, for an LI of 2.6.

Even this was a simple example. In baseball, you can have a whole set of possible events, with a whole other set of runner movements for each event. A single advances runners one or two bases, or causes some of them to be out. A ground out can advance runners. When we put the whole gamut of possibilities into the process, we get an LI of 2.9, whether using recipe #1 or #2.

Recipe #3—The Great Hitter

Another way to think of leverage is to consider the impact to winning if you bring in a great player to replace an average player. For example, if the game is a blowout, the win expectancy will remain relatively unchanged when Manny Ramirez hits in place of an average hitter. But, in the ninth inning illustration we’ve been considering, the home team on defense will certainly quiver seeing Manny come into the game, in the top of the ninth inning, with men on first and second, and no outs—Manny representing the tying run.

We’re going to define a hitter such that he adds about .010 wins each time he comes to bat, on average. This represents what a great hitter adds to his team. Some times, this great hitter will add .030 wins, some times .050 wins, and some times .002 wins. Overall, this great hitter will average +.010 wins per plate appearance.

Now, let’s go back to our crucual ninth inning situation. With an average hitter here, the home team has a .841 chance of winning (as we’ve already mentioned). With a great hitter at the plate, their chances of winning plummets to .812. This is a drop of .029 wins. Remember, we just said that a great hitter will add .010 wins at some random point in the game. In this particular point in the game, the leverage of the situation makes it so that he has much more impact. The calculation for the LI is .029 / .010 = 2.9. The result of this recipe is also consistent with the results of the earlier recipes.

What if instead of a great hitter, we put in a poor hitter? How would the LI change? If we put in someone who, on average, is worth .005 wins worse than average into this situation, the expected winning percentage for the defensive team jumps to .856, or a change of .015 wins, for an LI of 3.0. In fact, you can put in practically any hitter of any quality, and the LI calculation will come back with pretty much a number between 2.8 and 3.0.

That first column is simply an identifier for the situation. It corresponds to the x-axis in the graph below. The score is from the perspective of the home team (Cubs). The LI is the Leverage Index. The highest-leverage situation was when Lee was at bat, with the bases loaded, Cubs up by 2, and 1 out. The leverage index is a numerical descriptor for the crucialness of the situation. How tense was the game? It was at its most tense when Derrek Lee was at bat. And what happened? Lee delivered a double, scoring two runs to tie the game. The combination of the high leverage situation and a big outcome (the double) resulted in the biggest play of the game.

Here is that fateful inning, described through the eyes of Leverage Index and Win Expectancy.